curve #5326

y² + y = x³ − 576257221

a-invariants: [0, 0, 1, 0, -576257221]
c-invariants: [0, 497886238728]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 186707339523
discriminant (Δ): -143455270089534147603
Faltings height: 2.5466
naive height: 53.8673
primes of bad reduction: 3, 53, 523
regulator: 1705.881665039014467241318701248544390042655613
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:28 UTC
last updated: 2026-10-21 13:09:28 UTC

Witness: 4 independent points log in to add more points →

(4057, 257291)
(47593/4, 10381023/8)
(22489/25, 1539388/125)
(229597/49, 109705616/343)

Commentary

Cube-sum curve x^3 + y^3 = 249471 (Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found in an exhaustive scan of cube-free k <= 7*10^5 by minimal-model naive height, with a Mestre-Nagao a_p sieve (p = 1 mod 3) and PARI ellrank 2-descent; companion to #5313 (k = 111690).

last edited by David Renshaw at 2026-10-21 13:09:28 UTC · history